Textbook Question
23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.28
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
g(t) = 6√t
Verified step by step guidance1
Step 1: Recognize that the function g(t) = 6√t can be rewritten using exponent notation as g(t) = 6t^{1/2}.
Step 2: Apply the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n*t^{n-1}.
Step 3: Differentiate g(t) = 6t^{1/2} using the power rule. The derivative of t^{1/2} is (1/2)t^{-1/2}.
Step 4: Multiply the derivative of t^{1/2} by the constant 6, resulting in g'(t) = 6 * (1/2)t^{-1/2}.
Step 5: Simplify the expression to find the derivative g'(t) = 3t^{-1/2}.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Power Rule
The power rule is a basic technique for finding the derivative of functions of the form f(x) = x^n, where n is a real number. According to this rule, the derivative f'(x) is given by n*x^(n-1). This rule simplifies the differentiation process, especially for polynomial functions and can be applied to functions involving roots, such as √t.
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Square Root Function
The square root function, denoted as √x, is a specific type of function that returns the non-negative square root of x. In calculus, it is important to understand how to differentiate this function, as it can be expressed in terms of exponents (x^(1/2)). This transformation allows the application of the power rule to find its derivative effectively.
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Related Practice
Textbook Question
Reproduce the graph of f and then plot a graph of f' on the same axes. <IMAGE>
1
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Textbook Question
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^10x
Textbook Question
The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle θ when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)? <IMAGE>
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
s(t) = cos 2^t
Textbook Question
Find d/dx(ln(x/x²+1)) without using the Quotient Rule.